While no particular line or even finite set of lines is necessary, what is necessary is that both 1. That there is at least one line, and 2. For each line there is also a next longer line.

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WM claims to know how to map bijectively the set of infinite binary sequences, B, linearly to the set of reals and then map that image set of reals linearly ONTO the set of all paths, P, of a Complete Infinite Binary Tree.

But each binary rational in |R is necessarily the image of two sequences in B but that one rational can then only produce one image in P, so the mapping cannot be the bijection WM claims.